Nonstandard finite differences numerical methods for a vegetation reaction–diffusion model

In this work we derive NonStandard Finite Differences (NSFDs) (Anguelov and Lubuma, 2001; Mickens, 2020) numerical schemes to solve a model consisting of reaction–diffusion Partial Differential Equations (PDEs) that describes the coexistence of plant species in arid environments (Eigentler and Sherratt, 2019). The new methods are constructed by exploiting a-priori known properties of the exact solution, such as positivity and oscillating behavior in space. Furthermore, we extend the definition of NSFDs inspired by the Time-Accurate and High-Stable Explicit (TASE) (Bassenne et al., 2021) methodology, also exploring the existing connections between nonstandard methods and the Exponential-Fitting (EF) (Ixaru, 1997; Ixaru and Berghe, 2010) technique.